1. **State the problem:** Expand and simplify the expression $$(1 - 3d^4 + 4d^2)(d + 3)$$ and arrange the terms in descending powers of $d$. 2. **Recall the distributive property:** To expand, multiply each term in the first polynomial by each term in the second polynomial. 3. **Multiply each term:** - $1 \times d = d$ - $1 \times 3 = 3$ - $-3d^4 \times d = -3d^{5}$ - $-3d^4 \times 3 = -9d^{4}$ - $4d^2 \times d = 4d^{3}$ - $4d^2 \times 3 = 12d^{2}$ 4. **Write all terms together:** $$d + 3 - 3d^{5} - 9d^{4} + 4d^{3} + 12d^{2}$$ 5. **Arrange terms in descending powers of $d$:** $$-3d^{5} - 9d^{4} + 4d^{3} + 12d^{2} + d + 3$$ 6. **Final answer:** $$\boxed{-3d^{5} - 9d^{4} + 4d^{3} + 12d^{2} + d + 3}$$